This problem is foundational and important; moreover, it has a variety of applications, with the most typical example being the estimation of covariance matrices.
Let \(Y_1,... ,Y_n\)  be a sequence of random vectors; then. we can estimate its covariance matrix \(\Sigma = \mathbb {E}[Y_1 Y_1^\top ]\)  using the empirical mean \(n^{-1} \sum _{\ell = 1}^n M_\ell \)  by defining \(M_\ell = Y_\ell Y_\ell ^\top \) .
This setup can easily be applied for estimating Fisher information matrices, for example.
Other applications include estimation of adjacency matrices of random graphs [1]}, a signal recovery of the compressed sensing [2]}, and linear regression of overparameterization [3]}.
With the growing importance of data in the modern data science, it is increasingly becoming important to study the upper bound with various settings, such as high-dimensional and more complex distributions and dependencies.
